Given Abaddons attitudes towards loyalty and keeping black legion a unified force when he calls I would expect it to be a case of repainting your stuff black. Its also not a choice you get to back out of easily.

Atleast that's what can be gathered from the Black legion wiki/lexicanum , since I really don't like them enough to pay much attention to them normally.

Annoyingly, I wrote out a description of why it's a no, and why the question (to a Black Legionnaire) wouldn't ever come up...only for my browser to crash for no good reason. If you're interested, I don't mind rewriting it, but am a bit too annoyed to do that just now!

But again, in short, Abaddon gave the Black Legion three rules:

1. In all you do, seek to undermine and cause damage to the Imperium
2. Wear our colours and let them know you do this in our name
3. No matter where you are when it happens, when the Legion calls a black crusade, be there for it

However, as indicated, some of the god-specific warbands (and the Thousand Sons rubricae) have formed distinct warbands within the Black Legion that adopt a slightly distinctive visual identity - although they're still primarily black and gold.

Ask the Daemon Google about the Hounds of Abaddon, Sons of the Cyclops, Bringers of Decay or Children of Torment.

Then ask him about the Oath-Broken. That's where someone proposing to retain their old legion's colours would likely end up.

Edit: I calmed down. Scorching our armour black is an admission and acceptance of our failure to overthrow the Imperium. We also wear the Eye of Horus partly as a mark of respect to the fact that we are spiritual successors to the vision of the first Warmaster, but partly because it was once the Eye of Terra, and we claim ownership of Terra as we claim ownership of the Imperium and humanity's destiny.

The old bloodlines do not matter - they are merely defeat masquerading as defiance. The Black Legion is a host of brothers from all bloodlines, who have ceased fighting for a past we have already lost and thrown off the shackles of our failed fathers to seek a new bond of brotherhood with each other in pursuit of a common cause. For an astartes, joining the Black Legion is similar to joining the medieval crusading orders. It is done because you believe first and foremost in the ideals of the order, rather than because you simply desire the same end goal. For that, there are the wider armies of chaos, much as there were once myriad accompaniers on the original crusades.

As a result, the question of retaining your former legion's colours would never arise. You have made a decision to absolve yourself of your past, and embrace Abaddon's vision for the future. There would be no reason to want to keep the colours of your old legion in such a situation, and so the question never really arises.

Chaos Space Marines Black Legion Supplement, 6th Edition. pg. 25, boxed section Mark of Shame:He ordered his warriors to paint their armour black and strike all other symbols of their past allegiances... More important than a mark of shame, the black of the Legion's armour was part of Abbadon's plan to unify the Traitor Legions for his great war against the Imperium. So potent a symbol has it become that few other warbands wear black armour. It was a clear message that told of a Space Marine's devotion to Abaddon and that he had forsaken all other oaths.

Basically a sourcing of what everyone said above, but it leaves little doubt that when you join the Black Legion, you're in it 100%. Also, instead of a god mark, I would love to see a Mark of Shame for chaos undivided. Because it's so fluffy! And yes, I know Veterans of the Long War is basically the same thing, and chaos fanboys will get all up in arms about how "Mark of Shame" is a negative thing and really just Imperial propaganda, but it's in the book and it's fluffy and I like it.

You can keep your god, that's for sure, here's the officially dedicated chaos cults within the Black Legion.
Hounds of Abbadon: Khorne cult, have red stripes and shoulder guard to look cool
Children of Torment: Slaneesh cult, with purple patches
Sons of the Cyclops: Tzeentch, a big old sorcerer coven (with Rubricae!) Get some blue on that (noted for their excellent capes)
Bringers of Decay: Nurgle, with nurgle green stuff happening.

As for leaving your chaos god, I assume that's no different than normally trying to leave your chaos god. Maybe the Black Legion would shelter you a bit, but I doubt it. Have fun running from your former allies. And the mutations in your body...